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On critical exponent for existence of positive solutions for some semipositone problems involving the weight function. (English) Zbl 1175.35064

Summary: We study existence of positive solution for the semipositone problem of the form
\[ -\Delta u=\lambda m(x)u^\alpha-c, \quad x\in\Omega, \qquad u(x)=0, \quad x\in\partial\Omega, \]
where \(\Delta\) denote the Laplacian operator, \(\Omega\) is a smooth bounded domain in \(\mathbb R^N\) with \(\partial\Omega\) of class \(C^2\), \(\lambda\), \(c\) are positive parameters and the weight \(m(x)\) satisfying \(m(x)\in C(\Omega)\) and \(m(x)\geq m_0> 0\) for \(x\in\Omega\). A critical exponent is obtained for existence of positive solution by applying the method of sub-super solution.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B33 Critical exponents in context of PDEs
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